Is there any theorem from linear algebra about writing random variables as a linear combination of their elements?Suppose that X is a random variable which belongs to a standard probability space ( Ω , B , μ ). Could anybody provide some theorem and its details where X can be written as X = ∑ i = 1 k w i x i , where w i ≠ 0 and x i ∈ R ?

Name: Is there any theorem from linear algebra about writing random variables as a linear combination of their elements?Suppose that X is a random variable which belongs to a standard probability space ( Ω , B , μ ). Could anybody provide some theorem and its details where X can be written as X = ∑ i = 1 k w i x i , where w i ≠ 0 and x i ∈ R ?
Uploaded: 2022-02-23T17:22:57+00:00
Description: Is there any theorem from linear algebra about writing random variables as a linear combination of their elements?Suppose that X is a random variable which belongs to a standard probability space ( Ω , B , μ ). Could anybody provide some theorem and its details where X can be written as X = ∑ i = 1 k w i x i , where w i ≠ 0 and x i ∈ R ?

Question

Is there any theorem from linear algebra about writing random variables as a linear combination of their elements?Suppose that X is a random variable which belongs to a standard probability space   (  Ω  ,      B    ,  μ  ). Could anybody provide some theorem and its details where X can be written as   X  =      ∑          i      =      1        k        w    i        x    i  , where       w    i    ≠  0 and       x    i    ∈      R  ?

billyfcash5n · Accepted Answer

Solution:Sure, not a fancy theorem, but suppose X is a real-valued random variable with finite support   {      x    1    ,  .  .  .  ,      x    k    }. Then   X  =      ∑          i      =      1        k              1              X      =              x        i                  x    i    ..Notice the indicator weights are Bernoulli random variables.